Strong local passivity in finite quantum systems

Passive states of quantum systems are states from which no system energy can be extracted by any cyclic (unitary) process. Gibbs states of all temperatures are passive. Strong local (SL) passive states are defined to allow any general quantum operation, but the operation is required to be local, being applied only to a specific subsystem. Any mixture of eigenstates in a system-dependent neighborhood of a nondegenerate entangled ground state is found to be SL passive. In particular, Gibbs states are SL passive with respect to a subsystem only at or below a critical system-dependent temperature. SL passivity is associated in many-body systems with the presence of ground state entanglement in a way suggestive of collective quantum phenomena such as quantum phase transitions, superconductivity, and the quantum Hall effect. The presence of SL passivity is detailed for some simple spin systems where it is found that SL passivity is neither confined to systems of only a few particles nor limited to the near vicinity of the ground state.

Figure 1

Local energy ${\Omega }_{\circ }$ for $\kappa =2$ in $\mathcal{S}$ with system state $\rho$ parametrized by the probability differences ${\delta }_0$, ${\delta }_1$. Darker shading indicates greater local energy. ${\Omega }_{\circ }=0$ for any eigenmixture $\rho$ in a neighborhood of the ground state $|E_0\rangle$. Gibbs states are SL passive below the critical temperature $T_{*}$.

Figure 2

Critical temperatures $T_{*}$ below which ${\Omega }_{\circ }=0$ for selected coupling anisotropies $\gamma$. The points in the inset are coupling strengths $\kappa$ where $T_{*}=0$. The superposed curve in the inset is condition (25) for ground state degeneracy.

Figure 3

Gibbs states' critical temperatures $T_{*}$ for SL passivity and zero local energy in $N$-particle spin chains. Curves for $N=6$ and beyond are visually indistinguishable.